The goal of this research project is to clarify the relation between locally compact groups and their associated von Neumann and C*-algebras. I have three objectives.First, I want to understand general criteria for a locally compact group G implying that its group von Neumann algebra L(G) is a non-amenable factor. Ideally, I will find a characterisation of such groups.Second, I want to find C*-superrigid groups, that is discrete groups G such that any other discret group whose reduced C*-algebra is isomorphic with the one of G, is already isomorphic wiht G.Third, I want to investigate the fine structure of the group von Neumann algebras of discrete hyperbolic groups. Based on Popa's deformation/rigidity theory, my aim is to generalise a theorem of Voiculescu on the possible spectral type of MASAs in free group factors to MASAs in group von Neumann algberas of hyperbolic groups.This project is going to be excecuted in Kyoto University and the University of Münster, two of the most renowned places in operator algebras and geometric group theory.